# Properties

 Label 2601.2.a.bc Level $2601$ Weight $2$ Character orbit 2601.a Self dual yes Analytic conductor $20.769$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2601 = 3^{2} \cdot 17^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2601.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$20.7690895657$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{16})^+$$ Defining polynomial: $$x^{4} - 4 x^{2} + 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 51) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( -1 + \beta_{1} + \beta_{3} ) q^{5} + ( 2 - \beta_{1} ) q^{7} + ( -\beta_{1} + \beta_{3} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( -1 + \beta_{1} + \beta_{3} ) q^{5} + ( 2 - \beta_{1} ) q^{7} + ( -\beta_{1} + \beta_{3} ) q^{8} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{10} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{11} + ( -1 - \beta_{2} - 2 \beta_{3} ) q^{13} + ( -2 + 2 \beta_{1} - \beta_{2} ) q^{14} + ( -2 - 2 \beta_{2} ) q^{16} + ( -3 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{19} + ( 2 \beta_{1} - \beta_{2} ) q^{20} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{22} + ( -3 + \beta_{1} - 3 \beta_{3} ) q^{23} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{25} + ( -2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{26} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{28} + ( -2 + 3 \beta_{2} ) q^{29} + ( 2 + \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{31} + ( -2 \beta_{1} - 4 \beta_{3} ) q^{32} + ( -4 + 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{35} + ( 6 + \beta_{1} + 2 \beta_{2} ) q^{37} + ( -4 - 4 \beta_{1} - \beta_{3} ) q^{38} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{40} + ( -3 - \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{41} + ( 1 + 2 \beta_{1} - \beta_{2} + 4 \beta_{3} ) q^{43} + ( -2 - 2 \beta_{1} - \beta_{2} ) q^{44} + ( 2 - 3 \beta_{1} - 2 \beta_{2} ) q^{46} + ( -2 - 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{47} + ( -1 - 4 \beta_{1} + \beta_{2} ) q^{49} + ( -4 + 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{50} + ( -2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{52} + ( -2 + 4 \beta_{2} - 2 \beta_{3} ) q^{53} + ( -3 - 2 \beta_{1} - \beta_{2} ) q^{55} + ( 2 - 2 \beta_{1} + 2 \beta_{3} ) q^{56} + ( \beta_{1} + 3 \beta_{3} ) q^{58} + ( -6 + \beta_{1} + 2 \beta_{2} ) q^{59} + ( 6 - \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{61} + ( 2 - 2 \beta_{1} - 4 \beta_{3} ) q^{62} -2 \beta_{2} q^{64} + ( -3 - 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{65} + ( 2 + 2 \beta_{1} + \beta_{2} ) q^{67} + ( 6 - 6 \beta_{1} + 5 \beta_{2} - 2 \beta_{3} ) q^{70} + ( 6 - 2 \beta_{1} + 3 \beta_{2} ) q^{71} + ( 2 + 4 \beta_{1} - 4 \beta_{2} + 6 \beta_{3} ) q^{73} + ( 2 + 8 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{74} + ( -2 - 3 \beta_{2} - 4 \beta_{3} ) q^{76} -\beta_{3} q^{77} + ( 2 \beta_{2} - 3 \beta_{3} ) q^{79} + ( 2 - 6 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{80} + ( -2 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{82} + ( -2 - 4 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{83} + ( 4 + 6 \beta_{2} - \beta_{3} ) q^{86} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{88} + ( -4 + 2 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} ) q^{89} + ( -2 + 2 \beta_{1} - 3 \beta_{3} ) q^{91} + ( -2 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} ) q^{92} + ( -4 - 6 \beta_{1} - 4 \beta_{3} ) q^{94} + ( 3 - 3 \beta_{1} - 3 \beta_{2} - 5 \beta_{3} ) q^{95} + ( -4 + 3 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{97} + ( -8 - 4 \beta_{2} + \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{5} + 8q^{7} + O(q^{10})$$ $$4q - 4q^{5} + 8q^{7} + 8q^{10} - 4q^{11} - 4q^{13} - 8q^{14} - 8q^{16} - 12q^{19} - 8q^{22} - 12q^{23} - 8q^{29} + 8q^{31} - 16q^{35} + 24q^{37} - 16q^{38} - 12q^{41} + 4q^{43} - 8q^{44} + 8q^{46} - 8q^{47} - 4q^{49} - 16q^{50} - 8q^{52} - 8q^{53} - 12q^{55} + 8q^{56} - 24q^{59} + 24q^{61} + 8q^{62} - 12q^{65} + 8q^{67} + 24q^{70} + 24q^{71} + 8q^{73} + 8q^{74} - 8q^{76} + 8q^{80} - 8q^{82} - 8q^{83} + 16q^{86} - 16q^{89} - 8q^{91} - 16q^{94} + 12q^{95} - 16q^{97} - 32q^{98} + O(q^{100})$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.84776 −0.765367 0.765367 1.84776
−1.84776 0 1.41421 −3.61313 0 3.84776 1.08239 0 6.67619
1.2 −0.765367 0 −1.41421 0.0823922 0 2.76537 2.61313 0 −0.0630603
1.3 0.765367 0 −1.41421 −2.08239 0 1.23463 −2.61313 0 −1.59379
1.4 1.84776 0 1.41421 1.61313 0 0.152241 −1.08239 0 2.98067
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$17$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2601.2.a.bc 4
3.b odd 2 1 867.2.a.m 4
17.b even 2 1 2601.2.a.bd 4
17.e odd 16 2 153.2.l.e 8
51.c odd 2 1 867.2.a.n 4
51.f odd 4 2 867.2.d.e 8
51.g odd 8 2 867.2.e.h 8
51.g odd 8 2 867.2.e.i 8
51.i even 16 2 51.2.h.a 8
51.i even 16 2 867.2.h.b 8
51.i even 16 2 867.2.h.f 8
51.i even 16 2 867.2.h.g 8
204.t odd 16 2 816.2.bq.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.2.h.a 8 51.i even 16 2
153.2.l.e 8 17.e odd 16 2
816.2.bq.a 8 204.t odd 16 2
867.2.a.m 4 3.b odd 2 1
867.2.a.n 4 51.c odd 2 1
867.2.d.e 8 51.f odd 4 2
867.2.e.h 8 51.g odd 8 2
867.2.e.i 8 51.g odd 8 2
867.2.h.b 8 51.i even 16 2
867.2.h.f 8 51.i even 16 2
867.2.h.g 8 51.i even 16 2
2601.2.a.bc 4 1.a even 1 1 trivial
2601.2.a.bd 4 17.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2601))$$:

 $$T_{2}^{4} - 4 T_{2}^{2} + 2$$ $$T_{5}^{4} + 4 T_{5}^{3} - 2 T_{5}^{2} - 12 T_{5} + 1$$ $$T_{7}^{4} - 8 T_{7}^{3} + 20 T_{7}^{2} - 16 T_{7} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 - 4 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$1 - 12 T - 2 T^{2} + 4 T^{3} + T^{4}$$
$7$ $$2 - 16 T + 20 T^{2} - 8 T^{3} + T^{4}$$
$11$ $$1 - 4 T - 6 T^{2} + 4 T^{3} + T^{4}$$
$13$ $$-47 - 68 T - 14 T^{2} + 4 T^{3} + T^{4}$$
$17$ $$T^{4}$$
$19$ $$-367 - 172 T + 18 T^{2} + 12 T^{3} + T^{4}$$
$23$ $$-271 - 132 T + 14 T^{2} + 12 T^{3} + T^{4}$$
$29$ $$( -14 + 4 T + T^{2} )^{2}$$
$31$ $$632 + 192 T - 48 T^{2} - 8 T^{3} + T^{4}$$
$37$ $$706 - 640 T + 196 T^{2} - 24 T^{3} + T^{4}$$
$41$ $$1 - 4 T - 2 T^{2} + 12 T^{3} + T^{4}$$
$43$ $$1297 + 196 T - 78 T^{2} - 4 T^{3} + T^{4}$$
$47$ $$-752 - 608 T - 72 T^{2} + 8 T^{3} + T^{4}$$
$53$ $$496 - 160 T - 56 T^{2} + 8 T^{3} + T^{4}$$
$59$ $$514 + 608 T + 196 T^{2} + 24 T^{3} + T^{4}$$
$61$ $$-1054 + 112 T + 132 T^{2} - 24 T^{3} + T^{4}$$
$67$ $$4 + 16 T + 4 T^{2} - 8 T^{3} + T^{4}$$
$71$ $$68 - 336 T + 164 T^{2} - 24 T^{3} + T^{4}$$
$73$ $$752 + 1952 T - 248 T^{2} - 8 T^{3} + T^{4}$$
$79$ $$-62 + 144 T - 52 T^{2} + T^{4}$$
$83$ $$-272 - 416 T - 120 T^{2} + 8 T^{3} + T^{4}$$
$89$ $$-2558 - 896 T - 20 T^{2} + 16 T^{3} + T^{4}$$
$97$ $$-632 - 224 T + 40 T^{2} + 16 T^{3} + T^{4}$$